# Is there a way to correlate groups of variables?

Suppose I have two binary variables, $X_1,X_2\in \lbrace 0,1\rbrace$. I can compute Pearson's correlation between them

$$\text{Corr}(X_1,X_2) = \frac{\text{Cov}(X_1,X_2)}{\sigma_{X_1}\sigma_{X_2}}$$

Now, I have been exploring association rule learning. In this type of machine learning, I am able to pick out rules like $\lbrace \text{onion,bun} \rbrace\Rightarrow \lbrace \text{hamburger}\rbrace$.

So when I look at these two different mechanisms, I see the following:

If I have a strong, postivie Pearson correlation between two variables, then if $X_1 = 1$, it is also likely that $X_2 = 1$.

If a customer has purchased onions and buns, it is likely they will buy a hamburger patty too (which is binary, 1 for purchase, 0 for no purchase).

So both of these things allow me to make inferences about binary variables. But unlike Pearson correlation, association rule learning allows me to use multiple inputs.

# MY QUESTION

Is there a version of correlation that accomplishes something similar to Pearson correlation but for groups of variables?